Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 10; no. 13; p. 2260
Main Authors: Mkhatshwa, Musawenkhosi Patson, Khumalo, Melusi
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.07.2022
Subjects:
Accuracy
Algorithms
Approximation
Boundary conditions
Chebyshev approximation
Chebyshev–Gauss–Lobatto points
Collocation methods
Computing time
Derivatives
Differential equations
Elliptic functions
Kronecker tensor product
Mathematics
Methods
Numerical analysis
Partial differential equations
Polynomials
quasilinearization method
spectral method
Tensors
three-dimensional elliptic PDEs
trivariate Lagrange interpolating polynomial
Variables
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math10132260